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URL: https://oeis.org/A373007

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A373007
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x_5, n) )^2.
5
1, 125, 2179, 15997, 78101, 272375, 823495, 2047613, 4765465, 9762625, 19487051, 34857463, 62748349, 102936875, 170182079, 262094461, 410338385, 595683125, 893871379, 1249381697, 1794395605, 2435881375, 3404824919, 4461748727, 6101640601, 7843543625, 10422071947
OFFSET
1,2
LINKS
FORMULA
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, x_2, x_3, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_7(d).
From Amiram Eldar, May 25 2024: (Start)
Multiplicative with a(p^e) = (p^(7*e+7) - p^(7*e+2) + p^2 - 1)/(p^7-1).
Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(s-2).
Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = zeta(8)/zeta(6) = Pi^2/10 = 0.986960440108... . (End)
MATHEMATICA
f[p_, e_] := (p^(7*e+7) - p^(7*e+2) + p^2 - 1)/(p^7-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 27] (* Amiram Eldar, May 25 2024 *)
PROG
(PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^2*sigma(d, 7));
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 25 2024
STATUS
approved